CHAPTER 3: AC POWER ANALYSIS

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CHAPTER 3 AC POWER ANALYSIS

• Instantaneous Average Power
• Max. Average Power Transfer
• RMS Value Apparent Power
• Complex Power Power Factor Correction

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Instantaneous Power
Instantaneous power (in watts) the power at any
instant of time
Where
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Instantaneous Power
The instantaneous power p(t) entering a circuit
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Average Power
Average Power (in watts) the average of the
instantaneous power over one period
Where
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Average Power
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Example 1
Given that
Find the instantaneous power and average power
absorbed by the passive linear network.
Reference Alexander, Sadiku Chapter 11 - page
461
7
Exercise 1
Calculate the instantaneous power and average
power if
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Circuit Elements
(a) Resistors
In purely resistive circuit, v and i are in
phase. ?v ?i. Therefore ? 0.
The average power is only dissipated in a purely
resistive circuit. For a purely inductive and
capacitive, the average power is zero.
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Circuit Elements
(b) Inductors
In purely inductive circuit, v leads by 90o,
therefore ? 90o
(c) Capacitors
In purely capacitive circuit, I leads by 90o,
therefore ? - 90o
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Example 2
Find the average power supplies by the source and
the average power absorbed by the resistor.
Reference Alexander, Sadiku Chapter 11 - page
462
11
Exercise 2
Calculate the average power absorbed by the
resistor and inductor. Find the average power
supplies by the voltage source.
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Maximum Average Power Transfer
Finding the maximum average power transfer a)
circuit with a load b) the Thevenin equivalent
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Maximum Average Power Transfer
In rectangular form, Thevenin impedance and Load
impedance
For maximum average power transfer, the load
impedance ZL must be equal to the complex
conjugate of the Thevenin impedance ZTh
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Maximum Average Power Transfer
In a situation in which the load is purely real
or purely resistive load (XL0), the load
impedance (or resistance RL) is equal to the
magnitude of the Thevenin impedance.
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Example 3
Determine the load impedance ZL that maximizes
the average power drawn from the circuit of
figure below. Calculate the maximum average power.
Reference Alexander, Sadiku Chapter 11 - page
466
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Exercise 3
Find the load impedance ZL that absorbs the
maximum average power for the circuit of figure
below. Calculate the maximum average power.
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Exercise 4
In Figure below, the resistor RL is adjusted
until it absorbs the maximum average power.
Calculate RL and the maximum average power
absorbed by it.
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Effective or RMS Value
The effective value of a periodic current is the
dc current that delivers the same average power
to a resistor as the periodic current.
Finding the effective current a) ac circuit
b) dc circuit
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Effective or RMS Value
For any perodic function x(t) in general, the rms
value is given by
The effective value of a periodic signals is its
root mean square (rms) value.
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Effective or RMS Value
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Effective or RMS Value
The average power can be written as
The average power absorbed by resistor R can be
written as
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Example 4
Determine the rms value of the current waveform
in figure below. If the current is passed through
a 2O resistor, find the average power absorbed by
the resistor.
Reference Alexander, Sadiku Chapter 11 - page
469
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Exercise 5
Find the rms value of the full wave rectified
sine wave in figure below. Calculate the average
power dissipated in a 6O resistor.
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Apparent Power
The apparent power (in VA) is the product of the
rms value of voltage and current.
S is known as the apparent power.
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Power Factor
The power factor is the cosine of the phase
difference between voltage and current. It is
also the cosine of the angle of the load
impedance.
Power Factor
is Power Factor Angle
where
? pf is lagging if the current lags voltage
(inductive load) ? pf is leading if the current
leads voltage (capacitive load)
For purely resistive circuit, pf1. With
inductors and capacitors in the circuit, pf may
reduced to less than 1.
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Example 5
A series connected load draw a current when the
applied voltage is Find the apparent power and
the power factor of the load. Determine the
element values that form the series connected
load.
Reference Alexander, Sadiku Chapter 11 - page
472
27
Exercise 6
Calculate the power factor of the circuit below
as seen by the source. What is the average power
supplies by the source?
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Complex Power
Complex power (in VA) is the product of the rms
voltage phasor and the complex conjugate of the
rms current phasor. As a complex quantity, its
real part is real power P and its imaginary part
is reactive power Q.
Complex power
VA
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Complex Power
Apparent power
VA
Real power
W
Reactive power
VAR
Q 0 for resistive loads (unity power factor) Q
lt 0 for capacitive loads (leading power factor) Q
gt 0 for inductive loads (lagging power factor)
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Complex Power
Power triangle
Impedance triangle
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Complex Power
Power Triangle
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Example 6

• The voltage across a load is
• and the current through the element in the
  direction
• of the voltage drop is
• Find
• the complex and apparent powers
• the real and reactive powers
• the power factor and the load impedance

Reference Alexander, Sadiku Chapter 11 - page
475
33
Example 7

• A load Z draws 12kVA at a power factor of 0.856
• lagging from a 120 Vrms sinusoidal source.
• Calculate
• the average and reactive powers delivered to the
  load
• the peak current
• the load impedance

Reference Alexander, Sadiku Chapter 11 - page
476
34
Exercise 7

• A sinusoidal source supplies 10kVAR reactive
• power to load
• Determine
• the power factor
• the apparent power delivered to the load
• the peak voltage
  

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Power Factor Correction
The process of increasing the power factor
without altering the voltage or current to the
original load is known as power factor correction.
Most loads are inductive. A load power factor is
improved (to make closer to unity, pf1) by
installing a capacitor in parallel with the load.

• Original inductive load b) inductive load with
  improved
• power factor

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Power Factor Correction
Phasor diagram showing the effect of adding a
capacitor in parallel with the inductive load
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Power Factor Correction
Power triangle illustrating power factor
correction
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Power Factor Correction
Value of required shunt capacitance
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Example 8
When connected to a 120 V (rms), 60Hz power line,
a load absorbs 4kW at a lagging power factor of
0.8. Find the value of capacitance necessary to
raise the pf to 0.95.
Reference Alexander, Sadiku Chapter 11 - page
482
40
Exercise 8
Find the value of parallel capacitance needed to
correct a load of 140kVAR at 0.85 lagging pf to
unity pf. Assume that the load is supplied by a
110V (rms), 60Hz line.